# Properties

 Label 9.5.d.a Level 9 Weight 5 Character orbit 9.d Analytic conductor 0.930 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 9.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.930329667755$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.39400128.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3}$$ $$+ ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5}$$ $$+ ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6}$$ $$+ ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7}$$ $$+ ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8}$$ $$+ ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3}$$ $$+ ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5}$$ $$+ ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6}$$ $$+ ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7}$$ $$+ ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8}$$ $$+ ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9}$$ $$+ ( 2 - 8 \beta_{1} + 8 \beta_{2} - \beta_{4} - \beta_{5} ) q^{10}$$ $$+ ( 108 - 2 \beta_{2} + 54 \beta_{3} - 3 \beta_{4} ) q^{11}$$ $$+ ( -8 + 21 \beta_{1} + \beta_{2} - 114 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{12}$$ $$+ ( -25 - 13 \beta_{1} - 26 \beta_{2} - 20 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} ) q^{13}$$ $$+ ( -114 + 2 \beta_{1} + 135 \beta_{3} + 21 \beta_{5} ) q^{14}$$ $$+ ( -175 + 13 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{15}$$ $$+ ( -30 - 42 \beta_{1} - 21 \beta_{2} - 41 \beta_{3} + 15 \beta_{4} - 30 \beta_{5} ) q^{16}$$ $$+ ( -60 - 3 \beta_{1} - 3 \beta_{2} - 162 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{17}$$ $$+ ( 336 + 33 \beta_{1} + 3 \beta_{2} + 207 \beta_{3} - 24 \beta_{4} + 3 \beta_{5} ) q^{18}$$ $$+ ( -52 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{19}$$ $$+ ( 324 - 28 \beta_{2} + 162 \beta_{3} + 24 \beta_{4} ) q^{20}$$ $$+ ( -56 - 74 \beta_{1} + 19 \beta_{2} - 342 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} ) q^{21}$$ $$+ ( -62 + 65 \beta_{1} + 130 \beta_{2} - 60 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{22}$$ $$+ ( -69 + 29 \beta_{1} + 27 \beta_{3} - 42 \beta_{5} ) q^{23}$$ $$+ ( -82 + 16 \beta_{1} + 65 \beta_{2} + 219 \beta_{3} - 13 \beta_{4} + 38 \beta_{5} ) q^{24}$$ $$+ ( 70 + 2 \beta_{1} + \beta_{2} + 127 \beta_{3} - 35 \beta_{4} + 70 \beta_{5} ) q^{25}$$ $$+ ( -282 - 26 \beta_{1} - 26 \beta_{2} - 486 \beta_{3} + 39 \beta_{4} - 39 \beta_{5} ) q^{26}$$ $$+ ( 27 - 72 \beta_{1} - 162 \beta_{2} + 216 \beta_{3} + 63 \beta_{4} - 27 \beta_{5} ) q^{27}$$ $$+ ( 134 + 84 \beta_{1} - 84 \beta_{2} - 30 \beta_{4} - 30 \beta_{5} ) q^{28}$$ $$+ ( -108 + 127 \beta_{2} - 54 \beta_{3} - 63 \beta_{4} ) q^{29}$$ $$+ ( -168 - 36 \beta_{1} - 114 \beta_{2} + 144 \beta_{3} - 3 \beta_{4} + 21 \beta_{5} ) q^{30}$$ $$+ ( 467 - 55 \beta_{1} - 110 \beta_{2} + 421 \beta_{3} - 92 \beta_{4} + 46 \beta_{5} ) q^{31}$$ $$+ ( -96 - 129 \beta_{1} + 81 \beta_{3} - 15 \beta_{5} ) q^{32}$$ $$+ ( -338 + 83 \beta_{1} - 35 \beta_{2} - 507 \beta_{3} + 91 \beta_{4} - 53 \beta_{5} ) q^{33}$$ $$+ ( -6 - 30 \beta_{1} - 15 \beta_{2} - 189 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{34}$$ $$+ ( 315 + 155 \beta_{1} + 155 \beta_{2} + 702 \beta_{3} + 36 \beta_{4} - 36 \beta_{5} ) q^{35}$$ $$+ ( -672 + 108 \beta_{1} + 255 \beta_{2} - 801 \beta_{3} - 21 \beta_{4} + 84 \beta_{5} ) q^{36}$$ $$+ ( 128 - 126 \beta_{1} + 126 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} ) q^{37}$$ $$+ ( -270 - 115 \beta_{2} - 135 \beta_{3} + 27 \beta_{4} ) q^{38}$$ $$+ ( 683 + 165 \beta_{1} + 152 \beta_{2} + 717 \beta_{3} - 14 \beta_{4} - 10 \beta_{5} ) q^{39}$$ $$+ ( -536 - 10 \beta_{1} - 20 \beta_{2} - 492 \beta_{3} + 88 \beta_{4} - 44 \beta_{5} ) q^{40}$$ $$+ ( 891 + 86 \beta_{1} - 783 \beta_{3} + 108 \beta_{5} ) q^{41}$$ $$+ ( 1940 - 356 \beta_{1} - 226 \beta_{2} + 489 \beta_{3} - 112 \beta_{4} - 55 \beta_{5} ) q^{42}$$ $$+ ( -170 + 332 \beta_{1} + 166 \beta_{2} + 176 \beta_{3} + 85 \beta_{4} - 170 \beta_{5} ) q^{43}$$ $$+ ( 660 - 241 \beta_{1} - 241 \beta_{2} + 1026 \beta_{3} - 147 \beta_{4} + 147 \beta_{5} ) q^{44}$$ $$+ ( -525 - 177 \beta_{1} + 78 \beta_{2} - 18 \beta_{3} - 174 \beta_{4} - 57 \beta_{5} ) q^{45}$$ $$+ ( -832 - 128 \beta_{1} + 128 \beta_{2} + 29 \beta_{4} + 29 \beta_{5} ) q^{46}$$ $$+ ( -2430 - 311 \beta_{2} - 1215 \beta_{3} + 120 \beta_{4} ) q^{47}$$ $$+ ( -532 + 275 \beta_{1} + 50 \beta_{2} + 1431 \beta_{3} - 18 \beta_{4} - 31 \beta_{5} ) q^{48}$$ $$+ ( -1010 - 151 \beta_{1} - 302 \beta_{2} - 891 \beta_{3} + 238 \beta_{4} - 119 \beta_{5} ) q^{49}$$ $$+ ( 192 + 439 \beta_{1} - 189 \beta_{3} + 3 \beta_{5} ) q^{50}$$ $$+ ( 216 + 126 \beta_{1} + 81 \beta_{2} - 783 \beta_{3} - 171 \beta_{4} + 270 \beta_{5} ) q^{51}$$ $$+ ( 108 - 252 \beta_{1} - 126 \beta_{2} + 8 \beta_{3} - 54 \beta_{4} + 108 \beta_{5} ) q^{52}$$ $$+ ( 1602 - 198 \beta_{1} - 198 \beta_{2} + 3132 \beta_{3} - 36 \beta_{4} + 36 \beta_{5} ) q^{53}$$ $$+ ( -1908 + 180 \beta_{1} + 9 \beta_{2} - 3024 \beta_{3} + 252 \beta_{4} - 234 \beta_{5} ) q^{54}$$ $$+ ( 365 - 29 \beta_{1} + 29 \beta_{2} - 124 \beta_{4} - 124 \beta_{5} ) q^{55}$$ $$+ ( 432 + 718 \beta_{2} + 216 \beta_{3} - 84 \beta_{4} ) q^{56}$$ $$+ ( 1236 - 88 \beta_{1} + 81 \beta_{2} + 507 \beta_{3} - 7 \beta_{4} ) q^{57}$$ $$+ ( 2416 + 8 \beta_{1} + 16 \beta_{2} + 2289 \beta_{3} - 254 \beta_{4} + 127 \beta_{5} ) q^{58}$$ $$+ ( 933 - 802 \beta_{1} - 972 \beta_{3} - 39 \beta_{5} ) q^{59}$$ $$+ ( 944 + 238 \beta_{1} + 38 \beta_{2} + 228 \beta_{3} + 416 \beta_{4} - 214 \beta_{5} ) q^{60}$$ $$+ ( 58 - 166 \beta_{1} - 83 \beta_{2} - 1264 \beta_{3} - 29 \beta_{4} + 58 \beta_{5} ) q^{61}$$ $$+ ( -1596 + 1000 \beta_{1} + 1000 \beta_{2} - 2862 \beta_{3} + 165 \beta_{4} - 165 \beta_{5} ) q^{62}$$ $$+ ( 48 + 192 \beta_{1} - 303 \beta_{2} + 2619 \beta_{3} - 36 \beta_{4} + 462 \beta_{5} ) q^{63}$$ $$+ ( 2074 + 501 \beta_{1} - 501 \beta_{2} + 111 \beta_{4} + 111 \beta_{5} ) q^{64}$$ $$+ ( -1728 - 287 \beta_{2} - 864 \beta_{3} - 81 \beta_{4} ) q^{65}$$ $$+ ( -2202 - 987 \beta_{1} - 519 \beta_{2} - 189 \beta_{3} + 153 \beta_{4} + 48 \beta_{5} ) q^{66}$$ $$+ ( -745 + 554 \beta_{1} + 1108 \beta_{2} - 938 \beta_{3} - 386 \beta_{4} + 193 \beta_{5} ) q^{67}$$ $$+ ( -1380 - 123 \beta_{1} + 999 \beta_{3} - 381 \beta_{5} ) q^{68}$$ $$+ ( -2659 + 85 \beta_{1} + 386 \beta_{2} - 780 \beta_{3} - 40 \beta_{4} - 253 \beta_{5} ) q^{69}$$ $$+ ( 310 + 176 \beta_{1} + 88 \beta_{2} + 3471 \beta_{3} - 155 \beta_{4} + 310 \beta_{5} ) q^{70}$$ $$+ ( -1050 - 876 \beta_{1} - 876 \beta_{2} - 1188 \beta_{3} + 456 \beta_{4} - 456 \beta_{5} ) q^{71}$$ $$+ ( 1428 - 897 \beta_{1} - 804 \beta_{2} - 99 \beta_{3} - 66 \beta_{4} - 21 \beta_{5} ) q^{72}$$ $$+ ( -2734 + 297 \beta_{1} - 297 \beta_{2} - 27 \beta_{4} - 27 \beta_{5} ) q^{73}$$ $$+ ( 5724 - 610 \beta_{2} + 2862 \beta_{3} - 378 \beta_{4} ) q^{74}$$ $$+ ( 2744 - 241 \beta_{1} - 436 \beta_{2} - 1611 \beta_{3} - 102 \beta_{4} + 185 \beta_{5} ) q^{75}$$ $$+ ( -1248 + 43 \beta_{1} + 86 \beta_{2} - 1277 \beta_{3} - 58 \beta_{4} + 29 \beta_{5} ) q^{76}$$ $$+ ( 9 + 1283 \beta_{1} + 216 \beta_{3} + 225 \beta_{5} ) q^{77}$$ $$+ ( -100 + 412 \beta_{1} + 668 \beta_{2} + 3108 \beta_{3} - 139 \beta_{4} + 317 \beta_{5} ) q^{78}$$ $$+ ( -344 - 646 \beta_{1} - 323 \beta_{2} - 2083 \beta_{3} + 172 \beta_{4} - 344 \beta_{5} ) q^{79}$$ $$+ ( -2184 - 410 \beta_{1} - 410 \beta_{2} - 5076 \beta_{3} - 354 \beta_{4} + 354 \beta_{5} ) q^{80}$$ $$+ ( 4617 + 837 \beta_{1} + 648 \beta_{2} + 3807 \beta_{3} + 108 \beta_{4} - 405 \beta_{5} ) q^{81}$$ $$+ ( -1072 - 545 \beta_{1} + 545 \beta_{2} + 86 \beta_{4} + 86 \beta_{5} ) q^{82}$$ $$+ ( -54 + 1297 \beta_{2} - 27 \beta_{3} + 756 \beta_{4} ) q^{83}$$ $$+ ( -3318 + 938 \beta_{1} + 1440 \beta_{2} - 4650 \beta_{3} - 16 \beta_{4} - 342 \beta_{5} ) q^{84}$$ $$+ ( 2958 - 168 \beta_{1} - 336 \beta_{2} + 3456 \beta_{3} + 996 \beta_{4} - 498 \beta_{5} ) q^{85}$$ $$+ ( -3498 - 1087 \beta_{1} + 3996 \beta_{3} + 498 \beta_{5} ) q^{86}$$ $$+ ( -4712 - 628 \beta_{1} - 899 \beta_{2} - 2505 \beta_{3} - 296 \beta_{4} + 244 \beta_{5} ) q^{87}$$ $$+ ( -418 + 310 \beta_{1} + 155 \beta_{2} - 4983 \beta_{3} + 209 \beta_{4} - 418 \beta_{5} ) q^{88}$$ $$+ ( 3102 + 1470 \beta_{1} + 1470 \beta_{2} + 3996 \beta_{3} - 1104 \beta_{4} + 1104 \beta_{5} ) q^{89}$$ $$+ ( 3870 + 432 \beta_{1} + 414 \beta_{2} + 594 \beta_{3} - 333 \beta_{4} - 99 \beta_{5} ) q^{90}$$ $$+ ( 6227 - 421 \beta_{1} + 421 \beta_{2} - 218 \beta_{4} - 218 \beta_{5} ) q^{91}$$ $$+ ( 6588 - 1042 \beta_{2} + 3294 \beta_{3} + 288 \beta_{4} ) q^{92}$$ $$+ ( -235 + 1101 \beta_{1} - 64 \beta_{2} - 5736 \beta_{3} - 158 \beta_{4} - 145 \beta_{5} ) q^{93}$$ $$+ ( -6122 - 1264 \beta_{1} - 2528 \beta_{2} - 5811 \beta_{3} + 622 \beta_{4} - 311 \beta_{5} ) q^{94}$$ $$+ ( -1626 + 196 \beta_{1} + 1512 \beta_{3} - 114 \beta_{5} ) q^{95}$$ $$+ ( 522 + 27 \beta_{1} - 909 \beta_{2} + 2862 \beta_{3} - 225 \beta_{4} + 117 \beta_{5} ) q^{96}$$ $$+ ( 244 - 784 \beta_{1} - 392 \beta_{2} + 9383 \beta_{3} - 122 \beta_{4} + 244 \beta_{5} ) q^{97}$$ $$+ ( -2910 - 1509 \beta_{1} - 1509 \beta_{2} - 4914 \beta_{3} + 453 \beta_{4} - 453 \beta_{5} ) q^{98}$$ $$+ ( 984 + 141 \beta_{1} + 1542 \beta_{2} + 3771 \beta_{3} + 165 \beta_{4} + 3 \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 15q^{4}$$ $$\mathstrut -\mathstrut 12q^{5}$$ $$\mathstrut -\mathstrut 99q^{6}$$ $$\mathstrut +\mathstrut 12q^{7}$$ $$\mathstrut +\mathstrut 99q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 15q^{4}$$ $$\mathstrut -\mathstrut 12q^{5}$$ $$\mathstrut -\mathstrut 99q^{6}$$ $$\mathstrut +\mathstrut 12q^{7}$$ $$\mathstrut +\mathstrut 99q^{9}$$ $$\mathstrut -\mathstrut 36q^{10}$$ $$\mathstrut +\mathstrut 483q^{11}$$ $$\mathstrut +\mathstrut 330q^{12}$$ $$\mathstrut -\mathstrut 6q^{13}$$ $$\mathstrut -\mathstrut 1146q^{14}$$ $$\mathstrut -\mathstrut 1026q^{15}$$ $$\mathstrut +\mathstrut 15q^{16}$$ $$\mathstrut +\mathstrut 1404q^{18}$$ $$\mathstrut -\mathstrut 258q^{19}$$ $$\mathstrut +\mathstrut 1614q^{20}$$ $$\mathstrut +\mathstrut 480q^{21}$$ $$\mathstrut -\mathstrut 369q^{22}$$ $$\mathstrut -\mathstrut 282q^{23}$$ $$\mathstrut -\mathstrut 1449q^{24}$$ $$\mathstrut -\mathstrut 273q^{25}$$ $$\mathstrut +\mathstrut 54q^{27}$$ $$\mathstrut +\mathstrut 1308q^{28}$$ $$\mathstrut -\mathstrut 1056q^{29}$$ $$\mathstrut -\mathstrut 1278q^{30}$$ $$\mathstrut +\mathstrut 1290q^{31}$$ $$\mathstrut -\mathstrut 1161q^{32}$$ $$\mathstrut +\mathstrut 279q^{33}$$ $$\mathstrut +\mathstrut 513q^{34}$$ $$\mathstrut -\mathstrut 2385q^{36}$$ $$\mathstrut +\mathstrut 12q^{37}$$ $$\mathstrut -\mathstrut 789q^{38}$$ $$\mathstrut +\mathstrut 1974q^{39}$$ $$\mathstrut -\mathstrut 1314q^{40}$$ $$\mathstrut +\mathstrut 7629q^{41}$$ $$\mathstrut +\mathstrut 9612q^{42}$$ $$\mathstrut -\mathstrut 285q^{43}$$ $$\mathstrut -\mathstrut 4212q^{45}$$ $$\mathstrut -\mathstrut 5760q^{46}$$ $$\mathstrut -\mathstrut 9642q^{47}$$ $$\mathstrut -\mathstrut 6771q^{48}$$ $$\mathstrut -\mathstrut 1863q^{49}$$ $$\mathstrut +\mathstrut 3027q^{50}$$ $$\mathstrut +\mathstrut 2457q^{51}$$ $$\mathstrut -\mathstrut 240q^{52}$$ $$\mathstrut -\mathstrut 405q^{54}$$ $$\mathstrut +\mathstrut 2016q^{55}$$ $$\mathstrut -\mathstrut 462q^{56}$$ $$\mathstrut +\mathstrut 5367q^{57}$$ $$\mathstrut +\mathstrut 6462q^{58}$$ $$\mathstrut +\mathstrut 6225q^{59}$$ $$\mathstrut +\mathstrut 7470q^{60}$$ $$\mathstrut +\mathstrut 3630q^{61}$$ $$\mathstrut -\mathstrut 7578q^{63}$$ $$\mathstrut +\mathstrut 15450q^{64}$$ $$\mathstrut -\mathstrut 7158q^{65}$$ $$\mathstrut -\mathstrut 13734q^{66}$$ $$\mathstrut -\mathstrut 5055q^{67}$$ $$\mathstrut -\mathstrut 10503q^{68}$$ $$\mathstrut -\mathstrut 13878q^{69}$$ $$\mathstrut -\mathstrut 9684q^{70}$$ $$\mathstrut +\mathstrut 8451q^{72}$$ $$\mathstrut -\mathstrut 14622q^{73}$$ $$\mathstrut +\mathstrut 26454q^{74}$$ $$\mathstrut +\mathstrut 21021q^{75}$$ $$\mathstrut -\mathstrut 4047q^{76}$$ $$\mathstrut +\mathstrut 2580q^{77}$$ $$\mathstrut -\mathstrut 12060q^{78}$$ $$\mathstrut +\mathstrut 4764q^{79}$$ $$\mathstrut +\mathstrut 18387q^{81}$$ $$\mathstrut -\mathstrut 9702q^{82}$$ $$\mathstrut -\mathstrut 1866q^{83}$$ $$\mathstrut -\mathstrut 6486q^{84}$$ $$\mathstrut +\mathstrut 12366q^{85}$$ $$\mathstrut -\mathstrut 37731q^{86}$$ $$\mathstrut -\mathstrut 21564q^{87}$$ $$\mathstrut +\mathstrut 14787q^{88}$$ $$\mathstrut +\mathstrut 20790q^{90}$$ $$\mathstrut +\mathstrut 34836q^{91}$$ $$\mathstrut +\mathstrut 33636q^{92}$$ $$\mathstrut +\mathstrut 19254q^{93}$$ $$\mathstrut -\mathstrut 12708q^{94}$$ $$\mathstrut -\mathstrut 13362q^{95}$$ $$\mathstrut -\mathstrut 3672q^{96}$$ $$\mathstrut -\mathstrut 28959q^{97}$$ $$\mathstrut -\mathstrut 9126q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$x^{5}\mathstrut +\mathstrut$$ $$11$$ $$x^{4}\mathstrut +\mathstrut$$ $$14$$ $$x^{3}\mathstrut +\mathstrut$$ $$98$$ $$x^{2}\mathstrut +\mathstrut$$ $$20$$ $$x\mathstrut +\mathstrut$$ $$4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 11 \nu^{4} - 121 \nu^{3} + 98 \nu^{2} + 1118 \nu - 220$$$$)/1098$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 11 \nu^{4} + 121 \nu^{3} - 98 \nu^{2} + 529 \nu + 220$$$$)/549$$ $$\beta_{3}$$ $$=$$ $$($$$$55 \nu^{5} - 56 \nu^{4} + 616 \nu^{3} + 649 \nu^{2} + 5488 \nu + 22$$$$)/1098$$ $$\beta_{4}$$ $$=$$ $$($$$$373 \nu^{5} - 260 \nu^{4} + 3958 \nu^{3} + 6817 \nu^{2} + 37558 \nu + 15082$$$$)/1098$$ $$\beta_{5}$$ $$=$$ $$($$$$-406 \nu^{5} + 623 \nu^{4} - 4657 \nu^{3} - 3583 \nu^{2} - 36898 \nu + 6206$$$$)/1098$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$22$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$26$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$22$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$237$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$46$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$22$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$23$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$46$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$549$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$270$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$135$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$572$$$$)/3$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$1 + \beta_{3}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 1.89154 − 3.27625i −0.102534 + 0.177594i −1.28901 + 2.23263i 1.89154 + 3.27625i −0.102534 − 0.177594i −1.28901 − 2.23263i
−5.67463 3.27625i −1.11837 8.93024i 13.4676 + 23.3266i 10.2044 5.89150i −22.9114 + 54.3399i 26.6364 46.1356i 71.6534i −78.4985 + 19.9746i −77.2081
2.2 0.307601 + 0.177594i 8.32172 + 3.42768i −7.93692 13.7472i −30.0804 + 17.3669i 1.95104 + 2.53225i 15.6054 27.0294i 11.3212i 57.5020 + 57.0484i −12.3370
2.3 3.86703 + 2.23263i −8.70335 2.29167i 1.96929 + 3.41090i 13.8760 8.01130i −28.5397 28.2933i −36.2418 + 62.7727i 53.8574i 70.4965 + 39.8904i 71.5451
5.1 −5.67463 + 3.27625i −1.11837 + 8.93024i 13.4676 23.3266i 10.2044 + 5.89150i −22.9114 54.3399i 26.6364 + 46.1356i 71.6534i −78.4985 19.9746i −77.2081
5.2 0.307601 0.177594i 8.32172 3.42768i −7.93692 + 13.7472i −30.0804 17.3669i 1.95104 2.53225i 15.6054 + 27.0294i 11.3212i 57.5020 57.0484i −12.3370
5.3 3.86703 2.23263i −8.70335 + 2.29167i 1.96929 3.41090i 13.8760 + 8.01130i −28.5397 + 28.2933i −36.2418 62.7727i 53.8574i 70.4965 39.8904i 71.5451
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(9, [\chi])$$.